Unfortunately it won't make you happy, because the function simply maps its inputs to its output, resulting in a very large hash (but you said that's OK).
Not quite because (1) a hash (by definition) returns a fixed-size bit string, as in, the number it returns must never exceed a certain value/size, and (2) the balance wouldn't be cryptographically tied to the the private key, I could modify the balance of the "hashed" output and it would be undetected.
A hash does not need to return a fixed-size bit string (although in software applications it usually does). Show me a definition that says it must be fixed-size. A hash function
"number of vowels in the input" is an example of a hash function with no upper size limit. The
Wikipedia article says "one can use the datum itself as the hashed value".
You wanted assurance that the balance had not changed "from a reference". Since you can easily calculate the balance of the output of my hash function and compare it to the reference balance, it meets your challenge. :-)
But I will let you off the 10 BTC payment.